You can stay signed in to OneNote for iPad or iPhone indefinitely, but if you want to change accounts or are using a shared iPad or iPhone, you can sign out of OneNote.
In OneNote, tap the Settings icon located in the upper right.
On the Settings screen, under Account, tap the account name you'd like to sign out of.
Tap Sign Out.
Accept the notification that appears to confirm signing out.
Note: You can close any notebooks you're not using (or not signed into) if you don't want to see them in the list. From the notebooks list, press and hold the notebook name for the notebook you'd like to close until a checkmark appears, and then on the bottom left, tap the Close button.